How MIMO cross-layer design enables QoS while detecting non-cooperative nodes in wireless multi-hop networks

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How MIMO cross-layer design enables QoS while detecting non-cooperative nodes in wireless multi-hop

networks

Abderrezak Rachedi, Hakim Badis, Abderrahim Benslimane

To cite this version:

Abderrezak Rachedi, Hakim Badis, Abderrahim Benslimane. How MIMO cross-layer design enables

QoS while detecting non-cooperative nodes in wireless multi-hop networks. Journal of Network and

Computer Applications (JNCA), Elsevier, 2014, 46, pp.395-406. �10.1016/j.jnca.2014.07.011�. �hal-

01024263�

How MIMO cross-layer design enables QoS while detecting non-cooperative nodes in wireless multi-hop networks

Abderrezak Rachedi

, Hakim Badis

, Abderrahim Benslimane

a

University Paris-Est (UPEM), Gaspard Monge Computer Science Laboratory (LIGM-UMR 8049), 77454 Marne-la-Vallée, France

b

University of Avignon, Computer Science Laboratory of Avignon (LIA), 84911 Avignon cedex 9, France

Abstract

Wireless Multi-hop Networks (WMNs) are based on the cooperation between nodes. The non-cooperative (selfish) nodes can affect the quality of services (QoS) delivered by the network. The solutions proposed in literature are based on the monitoring mechanism to detect non-cooperative nodes. However, the monitoring mechanism has to tackle a significant false alarm rate. The origin of these issues is mainly related to the interferences and the costs of the monitoring mechanism. In WMNs based on Single-Input Single-Output (SISO) technology, the interferences at the monitor (detector) node can affect the assessment and the accuracy of the monitor node’s observation. In this paper, we use Multi-Input and Multi-Output (MIMO) technology to tackle these drawbacks and to perform the monitoring mechanism without affecting the QoS. We propose a new MAC protocol based on the well-known SPACE-MAC protocol, named MIMODog. The collision at the monitor node can be avoided by tuning the antennas’ weights.

Therefore, the signal coming from other nodes than the monitored one can be nullified. Thus, this solution allows an important improvement of the accuracy of the monitor node’s observation. Moreover, we propose a monitoring capacity analysis using graph theory particularly Conflict Graph (CG), and asymptotic study. We illustrate that the capacity consumed in the case of MIMODog is costly compared to SPACE-MAC, but the accuracy of the observation is better. We demonstrate that the number of monitor nodes is Θ(

nlnn

) for a MIMO network with randomly located nodes n, each equipped with M antennas. Indeed, numerical results illustrate that by using MIMODog, the network can have a constant improvement M on an asymptotic number of monitor nodes compared to SISO 802.11 DCF MAC.

Keywords: Multi-hop wireless networks, Non-cooperative nodes, MIMO, Monitoring mechanism, Conflict Graph, QoS.

1. Introduction

Wireless Multi-hop Networks (WMNs) are a set of nodes based on the cooperation aspect to form and manage a network.

The nodes in WMNs act as router and terminal at the same time and a lack of cooperation between them implies the absence of any network. That is why it is important to deal with the non cooperative or selfish nodes problem. The problem of selfish nodes is that they keep their energy to transmit and route their own packets. In other words, the selfish nodes refuse to route and forward the packets of other nodes. The question is why do nodes act selfishly and ignore the cooperation aspect? To answer this question, we investigate the motivation of nodes to adopt this misbehavior. First of all, the resources in WMNs are limited in terms of energy, bandwidth, etc. Then, the nodes try to increase their lifetime duration by reducing their energy consumption and the cost of the transmission operation is im- portant in terms of energy. Secondly, when the nodes route and forward the packets of other nodes, this increases the delay of their own packets transmission and reduces their own average

∗

Corresponding author

Email addresses: rachedi@u-pem.fr (Abderrezak Rachedi), badis@u-pem.fr (Hakim Badis),

abderrahim.benslimane@univ-avignon.fr (Abderrahim Benslimane)

throughput. Thus, this operation may be perceived by nodes as punishment and not as global network interest. In order to deal with this problem, many researchers focus on the monitoring mechanisms in order to detect the selfish nodes and to punish them [1, 2, 3, 4, 5]. However, all the proposed mechanisms are based on the classical SISO (Single-Input Single-Output) tech- nology to monitor the communication channel and to detect the non forwarding nodes (selfish behavior). The most cited mech- anism is Watchdog [1]. Many proposed solutions are based on it, but they suffer from a high false alarm rate. The main prob- lem of these mechanisms is related to the interference at the monitor (detector) node which makes the results of its obser- vations wrong. These mechanisms are mainly considering the IEEE 802.11b/g as MAC protocol, so as far as we know, we are the first authors using Multiple-Input Multiple-Output (MIMO) system with IEEE 802.11n to remove the problem of false ob- servation at the monitor nodes [6].

In this paper, we extend the study of our proposed MAC pro-

tocol called MIMODog [6] based on Multi-Input Multi-Output

(MIMO) technology, and particularly a SPACE-MAC proto-

col [7] to significantly reduce the potential interferences at the

monitor (detector) nodes and to enhance the accuracy of the

monitoring results. The most significant added value consists

in i) the modeling of MIMODog to evaluate the impact of the

monitoring process on the network performance, ii) the propo- sition of a monitoring capacity analysis based on graph theory particularly conflict graph, iii) an asymptotic study proposed to investigate lower and upper bounds of the number of monitor nodes.

The contributions of this paper can be summarized in five points:

• We illustrate that the monitoring problem persists in WMNs, even when we use MIMO technology;

• We propose a new MIMO MAC protocol called MI- MODog to reduce the interferences at the monitor (detec- tor) nodes without negatively impacting the total network capacity;

• We present and discuss a different impact on the moni- toring process with: DCF MAC, SPACE-MAC and MI- MODog. MIMODog significantly enhances the monitor- ing process without affecting the network capacity;

• We propose a monitoring capacity analysis using con- flict graph from graph theory for different MAC protocols:

DCF MAC, SPACE-MAC and MIMODog;

• We investigate lower and upper bounds of the number of monitor nodes based on asymptotic study for MIMODog.

Moreover, the obtained numerical results illustrate that the proposed solution overcomes the drawbacks of classical monitoring mechanisms.

This paper is organized as follows: an overview of coopera- tion models based on monitoring mechanisms and the SPACE- MAC protocol is given in section 2. Section 3 illustrates the DCF and SPACE-MAC protocols vulnerabilities in the moni- toring process. A new MAC protocol adapted to the monitoring process with more details on its design and its implementation is proposed in section 6. In addition, the monitoring capacity and its impact on wireless multi-hop networks is investigated in section 5. Moreover, the theoretical model in order to as- sess the asymptotic bound related to the monitor nodes number is presented in section 6. The numerical results are given and analysed. The final section concludes the paper and presents our future works.

2. Related Work

In this section, we present the existing works related to the non-cooperative nodes monitoring and the detection mod- els in Wireless Multi-hop Networks (WMNs), and we briefly overview the SPACE-MAC protocol [7].

2.1. Non-cooperative nodes monitoring and detection Models Two kinds of non-cooperative (selfish) nodes can be distin- guished: active and passive selfish nodes. The active selfish nodes, also called greedy nodes, try to get more resources by maliciously manipulating protocols’ parameters at MAC and routing layers. For instance, at the MAC layer with IEEE

802.11 DCF mode the active selfish nodes can manipulate the backoff parameters in order to quickly access the channel at the cost of their neighbour nodes [8]. However, the aim of passive selfish nodes is to optimize their benefits in terms of QoS (like throughput and delay) and minimize their costs like the energy consumption without maliciously manipulating protocols’ pa- rameters. The typical action of passive selfish nodes is to refuse to forward the packets of other nodes.

In this paper, we focus on the selfish behaviour of the poten- tial cooperative nodes, particularly passive selfish nodes. Co- operation is an important parameter in wireless multi-hop net- works, because without any packet forwarding, the ad hoc net- work cannot exist and the wireless coverage extension is not possible.

In literature, two main solutions were proposed to over- come the problem of selfish nodes. The first one is based on the reputation mechanisms which consist in assessing a node’s contribution like forwarding and routing functionalities [4, 1, 9, 10, 11, 12]. The reputation model called CONFIDANT is proposed to share the reputation metric and alarm messages in order to detect and punish the misbehaving nodes [9]. An- other model called CORE is proposed to implement the repu- tation function by using the monitoring technique. Each node computes the reputation value for every neighbour and refuses to provide services to misbehaving nodes when their reputation is lower than a certain threshold.

The second one is based on the economics mechanisms also called price or credit-based mechanisms. In these models the nodes are paid to offer message forwarding services and pay to receive forwarding services. The proposed incentive mod- els for cooperation are using the concept of virtual cash. For instance, many incentive models based on game theory are pro- posed [13, 14]. The nodes are rewarded for forwarding packets by trading virtual cash with source and next hop nodes [13].

Buttyan and Hubeaux [15] proposed nuglets as credits to man- age forwarding transactions. The source node pays relay in- termediate nodes by storing nuglets in the packet head. The intermediate nodes acquire the nuglets when they forward the packets. In [16] a hybrid model using the reputation and the price-based mechanism was proposed to overcome the issue of selfish nodes. However, all these solutions are based on the classical monitoring mechanism like Watchdog[4]. The Watch- dog mechanism consists in the overhearing of the channel ac- tivities in order to detect the non-forwarding (selfish) nodes, but it suffers from a high false positive rate. The main issue of the false observation is related to the collision problem at the monitor (detector) node. Despite, many models are proposed to enhance the false observation [5], but the problem persists. All these models consider the IEEE 802.11b/g as MAC protocol, so as far as we know, we are the first authors using Multiple-Input Multiple-Output (MIMO) system with IEEE 802.11n to remove the problem of false observation at the monitor nodes.

2.2. SPACE-MAC protocol: Spatial Reuse Using MIMO Channel-Aware MAC

The SPACE-MAC is a Media Access Control protocol for

networks with smart antennas which uses antenna weights to

schedule simultaneous transmissions on a single collision do- main. Antenna weights are exchanged via control packets (RTS and CTS

) [7].

The main contribution of SPACE-MAC is the fully dis- tributed MAC protocol that exploits the physical layer char- acteristics and cross-layer techniques to enable spatial reuse in scatter-rich multi-path environments. The main advantage of SPACE-MAC is that it enables multiple data streams at the same time in the same collision area, thereby increasing the overall capacity of the network. The channel control over- head introduced by channel estimation and beam coordination is minimal and effectively countered by the gain provided by the increase in the capacity of MIMO channels.

In SPACE-MAC, the first station that gains access to the channel determines the silence period. All other stations must remain idle following their transmission until the silence period is completed. In SPACE-MAC, the silence period is required because any station currently involved in the transmission is unaware of any other transmission that began during its data packet or acknowledgement packet transmission phase. Addi- tionally, any station that wishes to transmit must not interfere with this ongoing transmission and must not transmit if it can- not complete its entire packet exchange sequence before the end of the silence period.

Based on the RTS/CTS handshake of the existing transmis- sion and for each new communication in the same geographical vicinity, the new sender/receiver nodes will select their weights so that the signal from any existing communication node is nul- lified. This problem can be formulated as a quadratic optimiza- tion and reduced to an unconstrained optimization problem us- ing the null space method which in turn is an eigenvalue prob- lem. Any new additional transmission is only possible if both nodes of a same pair have enough degrees of freedom. For an M antenna system it can null out at most M − 1 stations de- pending on the environment. M is also known as the Degrees of Freedom (DOF). Every time a node nulls out another node, it consumes a DOF.

3. MAC protocols vulnerabilities in monitoring process In this section, we highlight vulnerabilities in the monitoring process in different cases: classical IEEE 802.11 DCF (with SISO system), SPACE-MAC (with MIMO system), and MI- MODog (with MIMO system). In this study, we focus on the origin of the mis-observation of the monitor (detector) nodes.

3.1. Case of 802.11 DCF MAC with single antenna

The monitoring process acts on the different network proto- col layers (eg. MAC, Routing). In this work, we focus on the network layer for the monitoring. The monitor node supervises the packet forwarding activities of its neighbor nodes and their packet integrity. Let us consider a small network illustrated in Figure 1.

1

RTS: Request to Send / CTS: Clear to Send

Figure 1: An ad hoc network scenario

Two simultaneous communications are possible: B-C and D- E. Node A acts as monitor and supervises the packet forward- ing activities of node B. When a node B forwards A’s packets to node C, the communication between D and E can create a colli- sion at node A and then directly impact the monitoring process.

Figure 2 depicts the monitoring problem based on 802.11 DCF MAC.

Figure 2: Monitoring problem based on 802.11 DCF MAC protocol

3.2. Case of SPACE-MAC with multiple antennas

In this subsection, we will show that the monitor nodes can- not recover collided packets using the standard SPACE-MAC protocol.

Let us consider the latest network (Figure 1). We assume that all nodes are silent at the beginning, i.e., there is no on- going communication and each node has 4 antennas. Node B wants to forward A’s packets to C and D wants to communi- cate with E. Node B transmits an RTS using the default weight vector, [1 1 1 1]/ √

4, or a random vector. The vector is nor-

malized to have an equal signal power regardless of the number

of antennas. The weight vector used to transmit the RTS will

be used to transmit the following data packet and to receive the

corresponding CTS and ACK. Once node C receives the RTS,

it responds with a CTS packet using the current weight vec-

tor. The weight vector used to transmit the CTS will be used

to receive the following data packet and to send an ACK. The

receiver estimates the SIMO (Single-Input Multi-Output) chan-

nel vector h

= w

H

, where w

is the weight vector of node

B and H

is M × M MIMO channel matrix with elements h

and the superscript H denotes an hermitian operation. In fact,

as there is no ongoing communication, nodes C (receiver) and

A (monitor) can switch their weight vectors to w

= h

and

w

= h

which maximize the combined channel and array

gain. When a node other than the designated receiver and the

neighbor monitor receives the RTS, say node K, it estimates the

effective channel H and adjusts the weight vector so that the

signal from the RTS sender is nullified (i.e., h

C

= 0) for the duration of time specified in the RTS duration field. When a node other than the sender of the RTS (B) receives the CTS, say node L, it estimates the effective channel and stores the weight vector for the duration specified in the CTS duration field. Af- ter the RTS/CTS handshaking, node B sends, C receives and A supervises a data frame respectively using the weight vectors w

, w

and w

chosen as described above.

Now let us say node D wants to initiate a transmission to- ward E. Since node D is not currently aware of the antenna weight used by node B (node D cannot overhear B’s RTS and C’s CTS), it cannot adjust its weight vectors meeting these con- ditions: w

H

w

= 0 (D’s signals cannot be nullified by A).

Consequently a collision will occur at node A. The example shown in Figure 3 describes such a process problem.

Figure 3: Monitoring problem based on SPACE-MAC protocol

4. MIMODog protocol

In order to avoid any interference at the monitor node, each new transmitting node must be aware not only of the weight vectors of the existing transmissions in the cover area, but also of the weight vectors used by the monitor nodes. To deal with this issue, we propose a new MIMO MAC protocol called MI- MODog. The basic idea is that the monitor nodes simulate a real reception by sending CTS packet control before starting their monitoring process. We use the previous example to illus- trate our MIMO MAC protocol functioning.

4.1. Basic protocol operation

When monitor node A hears an RTS packet from its forward- ing node B:

1. it estimates the SIMO channel vector h

= w

H

and switches its weight vector to w

= h

to well receive B’s packets for monitoring;

2. it sends a CTS packet after a SIFS time using a weight vec- tor ˆ w

meeting this condition: ˆ w

H

w

= 0 (the A’s CTS signal is nullified at B to avoid collisions with C’s CTS and to ensure that node B will not change its behaviour if it is malicious). The A’s CTS contains the weight vector w

and transmits using ˆ w

. The goal of this operation is to make all future transmitters in the neighborhood believe that node A will receive packets and that its weight vector w

should be considered.

Once it receives the CTS packet from A, each node should estimate the effective channel from A. Now, the transmission of D should ensure that the reception of A is not disturbed. So, it picks W

meeting w

H

w

= 0 before transmitting its RTS.

The process is graphically explained in Figure 4.

Figure 4: Monitoring mechanism with MIMODog

4.2. RTS/CTS Control Packet Format

In order to selectively tune in or tune out a particular trans- mission, the stations have to be aware of the antenna weights that are used by transmitting stations. This requires a mecha- nism to convey the antenna weights to all neighboring stations.

MIMODog uses RTS and CTS control packets to convey an- tenna weights. The proposed format for RTS and CTS control packets is shown in Figure 5. A separate s byte field is inserted in the payload of the RTS and CTS packets and stores M an- tenna element weights currently in use. A linear function f is used to obtain the value of s. For example, as each antenna weight can be a complex number, we can store them as a pair of real numbers occupying 4 bytes (per one complex number) and so f (M) = 4M. RTS and CTS packets are also used to per- form a channel estimation using pilot symbols embedded in the physical (PHY) preamble.

Figure 5: Access Control Packets

5. Monitoring capacity analysis

In this section, we investigate the monitoring capacity and its

impact on wireless multi-hop networks. The monitoring capac-

ity for cooperation can be evaluated by the ability of a monitor (detector) node to listen (overhear) its neighbors activities un- der interference consideration. In this regard, we focus on:

• the computation of the available capacity on the link be- tween the monitor-forwarding nodes;

• the necessary conditions to correctly perform monitoring.

5.1. Preliminary and definitions

The link capacity in a single-hop network can be defined as the physical transmission bit rate of the source, determined by:

the Shannon limit, the fixed modulation scheme and the bit er- ror rate. In a wireless multi-hop network (WMN) context, sev- eral links share the same transmission medium, and so the link capacity decreases when more simultaneous transmissions oc- cur. The sum of all active data flow throughputs in the same interference area gives the consumed capacity. Consequently, the available link capacity is the difference between the link capacity (in a single-hop network) and the consumed capacity (Figure 6):

Available Link Capacity = Link Capacity − Consumed Capacity.

(1)

Figure 6: Link Capacity.

Table 1 summarizes the notation used in the next sections.

A WMN can be seen as an undirected stochastic geometric graph, called connectivity graph G. Figure 7 shows an example of a connectivity graph. Node b acts as monitor node and node c as forwarding node.

We use the conflict graph to model the interference relation- ships between links and called it the Links Conflict Graph LCG.

Every link in connectivity graph G is represented by a node in conflict graph LCG. Two nodes in G are connected by an edge if the nodes corresponding to links in G cannot have simultaneous transmissions according to the protocol’s interference model.

For this purpose and as explained in [17], we use the following interference model: any link within distance H from (i, j) is a potential interfering link. This rule is called the distance-H interference model.

Figure 7: An example of a connectivity graph.

Figures 8.a and 8.b show distance-2 and distance-4 LCGs for the network topology presented in Figure 7. The link be- tween the monitor-forwarding nodes, 2 = (a, c), belongs to three cliques

in Figure 8.a and to one clique in Figure 8.b. We note that, as distance-H of the interference model increases, the number of cliques decreases.

1=(a,b)

2=(b,c) 4=(a,e)

5=(a,f)

6=(d,g) 7=(d,h) 3=(c,d)

1=(a,b)

3=(c,d) 2=(b,c)

6=(d,g) q1

q2

q3 5=(a,f)

4=(a,e)

7=(d,h)

q1

(a) LCG with H=2 (b) LCG with H=4

Figure 8: LCGs for network topology.

5.2. Case of 802.11 DCF MAC with single antenna

Based on LCG, the LCG-nodes in a maximal clique represent the maximal set of mutually contending wireless links, along which only one flow may transit at any given time, consequently only one LCG-node in a clique may be active. Accordingly, the sum of the rates of LCG-nodes in each maximal clique can- not exceed the capacity of the channel; these conditions are the clique constraints. Since the network must satisfy the capacity constraints for all cliques, we can write the clique constraints in a matrix form. We represent a set of flow rates as the column vector ˆ F of size n, where n is the number of links in the network G and ˆ F

is the average flow rate assigned to link i. Let C

be a column vector of size n with all entries equal to the channel capacity C

. Hence we have,

2

A clique in the conflict graph is a set of vertices that mutually conflict with

each other.

Table 1: Notation

Symbol Definition

G the connectivity graph

G-node a node in the connectivity graph G-link a link in the connectivity graph

LCG the links conflict graph

LCG-node a node in the link conflict graph LCG-link a link in the link conflict graph

C

the capacity on link i within a single-hop network F

(t) the instantaneous flow rate utilization on link i at time t

β the scaling factor

Q

the incidence matrix of link i Γ

the available capacity on link i

Γ

the available capacity on the link between the monitor-forwarding nodes

∀ i Q

.F ≤ C

. (2) where Q

is an incidence matrix, which is of order qxn. Here, q is the number of maximal cliques that this link i belongs to, and n is the total number of links. The union of the clique matrices across all the links gives the global clique matrix Q.

Note that the flow rate F

assigned to link i in an interval of time ]t − τ, t], can be written as:

F ˆ

= 1 τ

t

Z

t−τ

F

(r)dr, (3)

where F

(r) is the instantaneous flow rate utilization on link i at time r.

The clique constraints provide a necessary condition for a realizable schedule to exist, since there cannot be a feasible schedule over links that form a violated clique constraint. One might hope that these constraints would also be sufficient. Un- fortunately, that is only true for a special sub-class of graphs called Perfect Graphs [18]. In prior work [19], the authors have proved the sufficiency condition using clique constraints scaled by 0.46 (using a virtual conflict graph). In our work [17], we have generalized the notion of scaling factor, β, according to the used interference model. Clique constraints become:

∀ i Q

. F ˆ ≤ β.C

. (4) As the CG-node can be a part of multiple cliques, it consid- ers all the cliques that it belongs to, and takes the worst case available capacity over all the cliques. The available capacity on a CG-node , i, is

Γ

= min { (C

× β) − Q

F ˆ } . (5) Γ

is the available capacity on link i, taking into account ac- tive flows on i, as well as interference from neighboring links.

For example, in Figure 8.a, let the allocated flow on each LCG-node { 1 = (a, b), 2 = (b, c), 3 = (c, d), 4 = (a, e), 5 = (a, f ), 6 = (d, g), 7 = (d, h) } be denoted by { F ˆ

, F ˆ

, F ˆ

, F ˆ

, F ˆ

, F ˆ

, F ˆ

} . Then, the available capacity on the link between the monitor-forwarding nodes, 2 = (b, c), is:

Γ

= min



 

 

 

 

 

 

(C

× β) − ( ˆ F

+ F ˆ

+ F ˆ

) (C

× β) − ( ˆ F

+ F ˆ

+ F ˆ

+ F ˆ

) (C

× β) − ( ˆ F

+ F ˆ

+ F ˆ

+ F ˆ

) In the same way, from Figure 8.b:

Γ

= (C

× β) − ( ˆ F

+ F ˆ

+ F ˆ

+ F ˆ

+ F ˆ

+ F ˆ

+ F ˆ

).

When the forwarding node forwards the monitor node’s traf- fic, say θ, two conditions are necessary for a successful listening by the monitor node:

• The traffic from/to the monitor node should be null. Other- wise, the monitor node cannot hear the forwarding node’s activities. The capacity of the link between monitor- forwarding nodes becomes:



 



 



Γ

= min { (C

× β) − Q

F ˆ }

F

= 0 (6)

• The available capacity on the link between the monitor- forwarding nodes is more than the amount of the moni- tored traffic:

Γ

≥ θ. (7)

However, the previous conditions are not sufficient because a feasible schedule isn’t guaranteed.

5.3. Case of SPACE-MAC protocol with MIMO system

The SPACE-MAC protocol operates in M-dimensional

space, where M is the degrees of freedom (DoF) of the MIMO

channel. In this case, the interference relationship between dif- ferent links in the network are conserved in the same DoF. How- ever, the inter-DoF interference is null. Consequently, the ca- pacity of any MIMO link in the network is the sum of all ca- pacities on that link per DOF. Figure 9 illustrates the superposi- tion but independence of the link conflict graphs of the network topology shown in Figure 7. Each DoF has its own LCG. All the LGCs are similar if we apply the same interference model.

Figure 9: LCGs for network topology.

Let Γ

be the available capacity on link i under DoF i. The total available capacity on link i, Γ

, is given by the following formula:



 



 



Γ

= P

Γ

Γ

= min { (C

× β

) − Q

F ˆ

} (8) We assume that at any time slot t and under any DoF, a node can either transmit or receive, but not both. In other words, if a node transmits at time slot t and under DoF m, it cannot trans- mit or receive at the same time slot under the remaining DoFs.

To hear the forwarding node, the monitor node should use the same weight vector used by the forwarding node. This means that the monitor-forwarding pair uses the same DoF. Accord- ing to the DoF used by transmitter/receiver nodes located in the interference area of the monitor node, we distinguish two cases:

• If no transmitter/receiver node uses the same DoF as that used for monitoring, the monitor node can correctly hear the forwarding node’s activity. In this case, the link capac-

ity of the monitor-forwarding nodes is:



 

 

 

 

 

 

Γ

= P

Γ

Γ

= C

if j is the used DoF for monitoring Γ

= min { (C

× β) − Q

F ˆ } otherwise

(9)

• If there is at least one transmitter/receiver node using the same DoF as that used for monitoring, in this case and following the same reasoning as in section 5.2, two condi- tions are necessary for a successful listening by the moni- tor node:

– The traffic from/to the monitor node should be null.

The link capacity of the monitor-forwarding nodes becomes:



 



 



Γ

= min { (C

× β) − Q

F ˆ }

F

= 0 (10)

– The available capacity on the link between the monitor-forwarding nodes is more than the amount of the monitored traffic:

Γ

≥ θ. (11)

5.4. Case of MIMODog protocol with MIMO systems

The main goal of the MIMODog protocol is to avoid trans- missions and monitoring under the same DoF within the inter- ference area of the monitor node. Consequently, the link capac- ity of the monitor-relay nodes is similar to that of the SPACE- MAC protocol illustrated in equation 9. The resulting capacity is due to the reservation of the DoF for the monitoring process which demonstrates the advantage of MIMODog protocol. In- deed, to enable a successful monitoring regardless of the nodes activities in the interference area, a capacity reservation is nec- essary. The SPACE-MAC protocol is a DoF reservation mech- anism on the link between a source node and its 1-hop relay node. However, the MIMODog protocol introduces an addi- tional DoF reservation on the predecessor link.

6. Theoretical results of monitoring process

In this section, we present the obtained theoretical results re- lated to the monitoring and its impact on the network capacity.

6.1. Average monitor nodes’ number

Let G = (V , E) be a geometric graph with node set v and edge set E, where | V | = n and there is an edge between two nodes u and v with a probability P( k u − v k ). This probability expresses the relationship between the power of the received signal and the distance. If we apply the the log-normal shadowing model, P( k u − v k ) can be given as [20]:

P( k \ u − v k ) = 1 2

h 1 − erf

3.07 ln( k \ u − v k ) ξ

i , ξ , σ

η (12)

where:



 

 

 

 

 

 

k \ u − v k is the normalized distance

σ is the standard deviation of shadowing η is the path-loss exponent

In our work, we consider ξ = 0. In other words, we consider only the case where two nodes with distances less than the nor- malized distance 1 are connected. The mean number of hops is given by [20]:



 



 



hopcount =

node−degree

node − degree =

−1)

P

P

j=i+1

P( k x

− x

k ) (13) where m is the number of small squares containing at most one node.

Any traffic between two nodes in the network has at mean hopcount intermediate (forwarding) nodes and so hopcount po- tential monitor nodes. According to the number of source- destination pair, say K, the average monitor nodes’ number, N MN, is

N MN = k × hopcount. (14) Regardless of the used protocol (IEEE802.11 DCF, Space- MAC and MIMODog), equation 14 remains valid. Using equa- tion 14 and under different values of the number of nodes and degrees of adjacency, we obtain the results shown in figure 14.

In order to study the number of monitor (detector) nodes whatever the monitoring mechanism model used (SISO, MIMO, and MIMODog), we plot in figure 10 the number of monitor nodes (N MN) according to the network density and de- grees of neighborhood with different numbers of network traf- fic flows. We remark that the N MN increases not only with the network density, but also with the number of network traffic flows. However, the N MN significantly decreases when the de- gree of neighborhood increases. These results are coherent be- cause when the degree of neighborhood is significant the num- ber of potential relayer nodes also increases.

6.2. The impact of monitoring on the capacity

We assume that n nodes are randomly located on the surface of a square flat torus unit area. We use this geometric topology to avoid edge effects, which otherwise complicates the analysis.

Each node selects a destination randomly to which is sends λ(n) bits/s. We use slotted time for transmission.

The average length of of each source-destination pair in terms of links is hopcount + 1. The average required capacity over the entire network for λ(n) successful delivery is at least as follows:

• for 802.11 DCF MAC with single antenna: (hopcount + 1)nλ(n);

• for SPACE-MAC with M antennas:

M

. This

is because each node can use all its DoFs for spatial mul- tiplexing, and so the total required capacity is divided by the number of DoFs;

• for MIMODog protocol with M antennas:

M

.

As with the SPACE-MAC, the total required capacity is divided by the number of DoFs. However, to allow moni- toring without interference, MIMODog protocol extends the interference area of the forwarding node by adding the interference area of the monitor node (see section 4).

Consequently, the used protocol interference model be- comes distance-(H+1) from the forwarding node, rather than distance-H. In this case, the nodes in the interfer- ence area of the monitor node assume the impact of the distance-(H+1) as an increase in the number of hops.

In figure 11, we plot the average consumed capacity in the network according to the network density and the average hop count with different models of monitoring mechanism: SISO (M = 1), SPACE-MAC, and MIMODog (the number of anten- nas is set to 2). We remark that the required capacity in the case of MIMODog is more important than the case of SPACE-MAC particularly when the average hop count increases. However, the worst results are obtained in the case of SISO compared to both models based on MIMO technology.

In order to study the impact of the number of antennas (M), we plot in figure 12 the consumed capacity with different mon- itoring models. We remark that the gap between the consumed capacity in the case of MIMODog and SPACE-MAC remains constant when the number of antennas increases.

These results illustrate that the monitoring mechanism is costly if we want to reach 0% of misobservation, because MI- MODog consumes DoF resources to ensure the monitoring pro- cess.

6.3. Asymptotic study of the number of monitor nodes

In this subsection, we focus on the MIMODog asymptotic study of the lower and upper bound of monitor (detector) nodes’

number.

We consider a random multi-hop MIMO ad hoc network with n nodes, where each node, equipped with M antennas, is randomly located in a unit square area. Each node acts as a source node and transmits data to a randomly chosen destination node. The per-node throughput λ(n) is defined as the minimum data rate that can be sent from each source to its destination via multi-hop routing. The maximum data rate that a single data stream can support is W . We assume that a node’s transmitter is limited to a transmission range r(n). When a source node cannot transmit data to its destination node in one hop, multi-hop routing is needed to relay the data. Each node also has an interference range (1 + ∆)r(n), where ∆ is non negative-constant.

6.3.1. Lower bound of the number of monitor nodes

In [21], Gupta and Kumar showed that a capacity lower bound for a single-antenna ad hoc network is Ω(

nlnn

) by con- structing a feasible routing and scheduling scheme. Thus, by adopting the same routing and scheduling scheme in our MI- MODodog ad hoc networks as in [21], a number of monitor nodes lower bound of Ω(

nlnn

) can be obtained.

10 20 30

40 50 60

70 80 90

100 10 20 30 40 50 60 70 80 90 100 0

20 40 60 80 100 120

NMN

1 source node 5 source nodes 10 source nodes

Number of Nodes n

Average of node degree NMN

0 20 40 60 80 100 120

colour gradient

Figure 10: The number of monitor nodes (N MN) versus network density with different degrees of neighborhood.

0 1

2 3

4 5

6 7 8 10

20 30 40 50 60 70 80 90 100 0

500 1000 1500 2000 2500 3000 3500 4000

The average consumed capacity (Mbits/s)

SPACE-MAC with 2 antennas MIMODog with 2 antennas 802.11 DCF with single antenna

Average hopcount

Number of nodes n

The average consumed capacity (Mbits/s)

0 500 1000 1500 2000 2500 3000 3500 4000

colour gradient

Figure 11: The average consumed capacity in the network according to the network density and the average of hop count with λ(n) = 4 Mbits/s and M = 2.

6.3.2. Upper bound of the number of monitor nodes

As shown in [22], we partition the unit square of the network into small squares, with the size of each small square being cleverly chosen so that the maximum data rate that can be re- ceived by the nodes inside the small square can be accurately computed. This method enables to estimate the number of mon- itor nodes.

Lemma 1. For a square with a side length 1/ ⌈

√2

∆.r(n)

⌉ , there are

at most M − 1 times larger monitor nodes based on MIMODog than with SISO 802.11 DCF MAC.

Proof. Based on [22], for a square with a side length 1/ ⌈

√2

∆.r(n)

⌉ (as shown in Figure 13), the maximum number of total data streams that can be received by the nodes inside the square at any time slot for any routing scheme is not greater than M re- gardless of the number of receiving nodes inside the square.

Using our MIMODog, the presence of a monitor node in the

0 1

2 3

4 5

6 7 8 10

20 30 40 50 60 70 80 90 100 0

500 1000 1500 2000 2500 3000 3500 4000

The average consumed capacity (Mbits/s)

SPACE-MAC with 4 antennas MIMODog with 4 antennas 802.11 DCF with single antenna

Average hopcount

Number of nodes n

The average consumed capacity (Mbits/s)

0 500 1000 1500 2000 2500 3000 3500 4000

colour gradient

Figure 12: The average consumed capacity in the network according to the network density and the average of hop count with λ(n) = 4 Mbits/s and M = 4.

Figure 13: The receivers in a square with side length

square area consumes exactly one DOF. Let P

(t, i) be the prob- ability that node i is a monitor at time t in a square j. The number of monitor nodes in square j is given by P

i=1

P

(t, i).

Consequently, the new maximum number of total data streams that can be received by nodes inside a square j is not greater than M − P

i=1

P

(t, i) (M ≥ P

i=1

P

(t, i)).

In the same square and using SISO systems, the maximum number of total data streams that can be received by nodes in- side the square is 1. Only one monitor node can be functioning properly. So, there are at most M − 1 times fewer monitor nodes with SISO 802.11 DCF than with MIMODog.

Based on Lemma 1, we can now compute the maximum

number of monitor nodes that can be supported in the unit square network by taking the sum of the number of monitor nodes among all small squares.

Theorem 1. For a random multi-hop MIMO ad hoc network, a number of monitor nodes upper bound for all possible rout- ing and scheduling schemes is O(

nlnn

) with a high probability when n → ∞ .

Proof. The proof is similar to the proof of Theorem 1 in [22].

We can easily obtain this equation:

NMN ≤ 2 M √ π

∆

D √

n ln n + 2 √ 2 M

∆Dn + M √ ln n) Dn √

πn = O( M

√ n ln n ), (15) where N MN is the number of monitor nodes and D is the aver- age length of source-destination lines.

Combining the lower and upper bounds of the number of monitor nodes, we can see that the number of monitor nodes in a random multi-hop MIMO ad hoc network with n nodes is Θ(

).

6.3.3. Numerical results

By running 1000 instances, we obtain the average length of source-destination lines D = 0.52 (see [22]). We set ∆ = 1.

Using equation 1 and under different values of M = 1, 2, 3, 4 we obtain the results shown in figure 14. With M = 1 antenna, MIMODog is exactly the 802.11 DCF MAC. We can extract 2 elements :

• when the number of used antennas increases, the number

of monitor nodes increases,

0 2 4 6 8 10 12 14

10 20 30 40 50 60 70 80 90 100

Upper Bound of the number of monitor nodes

number of nodes n

1 Antenna: M=1 2 Antennas: M=2 3 Antennas: M=3 4 Antennas: M=4

Figure 14: The upper bounds of the number of the monitor nodes versus the number of nodes

• when the number of nodes increases, the upper bound of monitor nodes decreases. This is explained by the fact that the network is more and more dense with a high multi- hop connectivity and so MIMO interference cancellation is limited.

7. Conclusion

In this paper, we propose a new scheme called MIMODog based on SPACE-MAC protocol for exploiting adaptive an- tenna arrays in wireless multi-hop networks using a multi- party propagation channel to efficiently detect the selfish (non- cooperative) nodes. The proposed scheme nullifies the beam to competing nodes to enable concurrent transmissions and mon- itor in the same collision domain. Moreover, we highlight and discuss the different negative impacts on the monitoring process particularly the origin of the false alarm (misobservation) with:

DCF MAC, SPACE-MAC and MIMODog. We have shown that the required capacity in the case of MIMODog is more impor- tant (but remains close) than the case of SPACE-MAC, due to the allocation of some DoF resources for monitoring. In ad- dition, we have investigated the monitoring process capacity by using conflict graph model. The number of monitor nodes scaling laws for MIMO ad hoc networks with M antennas is evaluated by using asymptotic study. We have shown that the number of monitor nodes is at most M − 1 times larger based on MIMODog than with SISO 802.11 DCF MAC.

In our future works, we plan to evaluate the proposed solu- tion by extensive simulations with different parameters like the density of selfish nodes, mobility models, and traffic models.

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